Metrisability of two-dimensional projective structures

TL;DR

This paper constructs explicit local obstructions to determining when a two-dimensional projective structure admits a Levi-Civita connection, using invariants derived from second order ODEs and projective geometry.

Contribution

It provides the first explicit order-5 local obstruction for the metrisability of 2D projective structures and characterizes conditions for metric existence.

Findings

Obstruction is of order 5 in connection components.

Vanishing of the obstruction implies metric existence in real analytic case.

In generic cases, two order-6 invariants determine metrisability.

Abstract

We carry out the programme of R. Liouville \cite{Liouville} to construct an explicit local obstruction to the existence of a Levi--Civita connection within a given projective structure $[\Gamma]$ on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of $[\Gamma]$ or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.